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DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

Part of: Additive number theory; partitions Harmonic analysis in several variables

Published online by Cambridge University Press:  10 December 2009

Hyunsuk Kang  and
Alexander V. Sobolev
Hyunsuk Kang
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. (email: kang.maths@gmail.com)
Alexander V. Sobolev
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. (email: avsobolev@math.ucl.ac.uk)
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Abstract

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

MSC classification

Secondary: 11P21: Lattice points in specified regions 42B05: Fourier series and coefficients
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 118 - 134
Copyright
Copyright © University College London 2010

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References

[1]Bleher, P. and Bourgain, J., Distribution of the error term for the number of lattice points inside a shifted ball. In Analytic Number Theory, Vol. 1 (Allerton Park, IL, 1995) (Progress in Mathematics 3), Birkhäuser (Boston, 1996), 141–153.CrossRefGoogle Scholar
[2]Bleher, P. M., Cheng, Z., Dyson, F. J. and Lebowitz, J. L., Distribution of the error term for the number of lattice points inside a shifted circle. Comm. Math. Phys. 154(3) (1993), 433–469.CrossRefGoogle Scholar
[3]Bleher, P. and Dyson, F., Mean square limit for lattice points in a sphere. Acta Math. 67(3) (1994), 461–481.Google Scholar
[4]Bleher, P. and Dyson, F., Mean square value of exponential sums related to representation of integers as sum of two squares. Acta Math. 68(3) (1994), 71–84.Google Scholar
[5]Bleher, P. and Lebowitz, J. L., Variance of number of lattice points in random narrow elliptic strip. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), 27–58.Google Scholar
[6]Heath-Brown, D. R., The distribution and moments of the error term in Dirichlet divisor problem. Acta Arith. 60 (1992), 389–415.Google Scholar
[7]Hughes, C. P. and Rudnick, Z., On the distribution of lattice points in thin annuli. Int. Math. Res. Not. 13 (2004), 637–658.CrossRefGoogle Scholar
[8]Jarnik, V., Über die Mittelwertsätze der Gitterpunktlehre. V. Časopis Pěst. Mat. Fys. 69 (1940), 148–174.Google Scholar
[9]Marklof, J., The Berry–Tabor conjecture. In Proceedings of the 3rd European Congress of Mathematics (Barcelona, 2000) (Progress in Mathematics 202), Birkhäuser (Basel, 2001), 421–427.CrossRefGoogle Scholar
[10]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms, II. Duke Math. J. 115 (2002), 409–434.CrossRefGoogle Scholar
[11]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math. (2) 158 (2003), 419–471.CrossRefGoogle Scholar
[12]Marklof, J., Mean square value of exponential sums related to the representation of integers as sums of squares. Acta Arith. 117 (2005), 353–370.CrossRefGoogle Scholar
[13]Schmidt, W. M., Approximation to algebraic numbers. Enseign. Math. (2) 17 (1971), 187–253.Google Scholar
[14]Wigman, I., The distribution of lattice points in elliptic annuli. Q. J. Math. 57 (2006), 395–423.CrossRefGoogle Scholar